eigenvector

eigenvector

[′ī·gən‚vek·tər]

(mathematics)

A nonzero vector v whose direction is not changed by a given linear transformation T ; that is, T (v) = λ v for some scalar λ. Also known as characteristic vector.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

eigenvector

(mathematics)

A vector which, when acted on by a particular linear transformation, produces a scalar multiple of the original vector. The scalar in question is called the eigenvalue corresponding to this eigenvector.

It should be noted that "vector" here means "element of a vector space" which can include many mathematical entities. Ordinary vectors are elements of a vector space, and multiplication by a matrix is a linear transformation on them; smooth functions "are vectors", and many partial differential operators are linear transformations on the space of such functions; quantum-mechanical states "are vectors", and observables are linear transformations on the state space.

An important theorem says, roughly, that certain linear transformations have enough eigenvectors that they form a basis of the whole vector states. This is why Fourier analysis works, and why in quantum mechanics every state is a superposition of eigenstates of observables.

An eigenvector is a (representative member of a) fixed point of the map on the projective plane induced by a linear map.

This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Eigenvector

 

(or characteristic vector). An eigenvector of a linear transformation is a vector that does not change direction under the transformation and is simply multiplied by a scalar. For example, the eigenvectors of a transformation composed of rotations about some axis and of contraction toward the plane perpendicular to the axis are vectors directed along the axis.

The coordinates x₁x₂,..., x_n of the eigenvectors of a transformation of n-dimensional space with the matrix ║a_ik║ satisfy the system of homogeneous linear equations

where λ is an eigenvalue of the matrix. If the matrix of a transformation is Hermitian, then the eigenvectors are mutually perpendicular. As a result of a Hermitian transformation, a sphere becomes an ellipsoid whose major axes are eigenvectors of the transformation.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.